A diffusive plankton system with time delay and habitat complexity effects under Neumann boundary condition

Abstract

In this paper, we establish a delayed semilinear plankton system with habitat complexity effect and Neumann boundary condition. Firstly, by using the eigenvalue method and geometric criterion, the stability of the equilibria and some conditions for determining the existence of Hopf bifurcation are studied. Through analyzing the stability of positive equilibrium, we found that at the positive equilibrium the system may switch finitely many times from stable to unstable, then from unstable to stable, finally becoming unstable, i.e., the time delay induces a “stability switch” phenomenon. Secondly, the properties of Hopf bifurcation are derived by applying the normal form method and center manifold theory, including the bifurcation direction and the stability of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the theoretical results, and a biological explanation is given.